Prove Matrix Representations of p & x Don't Satisfy [-ih/2pi]

[bon]

Homework Statement

By taking the trace of both sides prove that there are no finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p,x] = -ih/2pi

Why does this argument fail if the matrices are infinite dimensional?

Homework Equations

The Attempt at a Solution

No idea really..

So I am guessing it'll be something like

[p,x] = px-xp

Let px-xp= some matrix C

Take trace of both sides,

Tr(px-xp) = Tr (C)

0=Tr(C)..

But don't see if/how this answers the question/what would be different in infinite case?

Thanks!

 

[fzero]

You also need to note that in this finite dim representation

C = - i\hbar I

where I is the identity matrix.

 

[bon]

Oh i see. Thanks

so i have Tr(-ih/2pi I)=0 so assuming p and x are nxn, we have (-ih)^n Tr(I) = n(-ih)^n = 0 which implies n = 0?

But how do things change if it is infinite?

 

[fzero]

Oh i see. Thanks

so i have Tr(-ih/2pi I)=0 so assuming p and x are nxn, we have (-ih)^n Tr(I) = n(-ih)^n = 0 which implies n = 0?

Is n=0 consistent with actually having a matrix representation?

But how do things change if it is infinite?

Is the trace operation well-defined for infinite-dimensional matrices?

 

[bon]

umm well I am not sure..can't it just be the sum to infinity rather than to some finite limit?

 

[fzero]

umm well I am not sure..can't it just be the sum to infinity rather than to some finite limit?

What about Tr I? Is that well-defined?

 

[bon]

it's infinity? oh i see..we run into problems? is that all that needs to be said?

Thanks!

 

[fzero]

Well you have similar problems trying to define Tr(xp) and Tr(px), so it's hard to define the difference between those traces as well.